Assuming thatr is a primitive root of the odd prime p, establish the following fact

a) the congruence r^((p-1)/2) =-1(mod p) holds.
b) if r' is any other primitive root of p , then rr' is not primitive root of p.
c) if the integer r' is such that rr'=1(mod p) then r' is primitive root of p.

and thnx alot

Dec 15th 2007, 02:58 PM

ThePerfectHacker

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Originally Posted by midosoft

a) the congruence r^((p-1)/2) =-1(mod p) holds.

We know that means since is odd we can write . Now it cannot be because . Thus, the only possibility is that .

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b) if r' is any other primitive root of p , then rr' is not primitive root of p.

We can write where . For be a primitive root it is necessary and sufficient that . Now , now for to be a primitive root it is necessary and sufficient that . But that is impossible because is an even number and among one is even by pigeonholing. Thus it impossible for to be a primitive root canal.

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c) if the integer r' is such that rr'=1(mod p) then r' is primitive root of p.

Now, because , thus, that means we can write for . We have that by hypothesis. That means because the order of is and . Thus , so and it means it is a primitive root.